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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8352, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,14,10]))
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pari:[g,chi] = znchar(Mod(4823,8352))
\(\chi_{8352}(71,\cdot)\)
\(\chi_{8352}(647,\cdot)\)
\(\chi_{8352}(1079,\cdot)\)
\(\chi_{8352}(1223,\cdot)\)
\(\chi_{8352}(1367,\cdot)\)
\(\chi_{8352}(2951,\cdot)\)
\(\chi_{8352}(4247,\cdot)\)
\(\chi_{8352}(4823,\cdot)\)
\(\chi_{8352}(5255,\cdot)\)
\(\chi_{8352}(5399,\cdot)\)
\(\chi_{8352}(5543,\cdot)\)
\(\chi_{8352}(7127,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((1567,5221,929,4033)\) → \((-1,-i,-1,e\left(\frac{5}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 8352 }(4823, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) |
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sage:chi.jacobi_sum(n)
